Chapter 2, Production Video Solutions, Competitive Equilibrium: Theory and Applications

Problem 1

Find $\operatorname{sp} S$, aff $S$, and $\operatorname{co} S$ for each of the following subsets of $\mathbf{R}^2$. Illustrate your answers with a diagram.
(a) $S=\{(1,1)\}$
(b) $S=\{(1,1),(2,2)\}$
(c) $S=\{(1,1),(2,3)\}$
(d) $S=\{(1,1),(2,3),(0,-1)\}$
(e) $S=\{(1,1),(2,3),(-1,0)\}$.

Laurie Huffman

Numerade Educator

04:16

Problem 2

Properly speaking, the equivalent definitions of the span, affine hull, and convex hull given in the text should be proved to be equivalent. To see how this is done, take as a definition that the convex hull of $S$ is the set of all convex combinations of the vectors in $S$. If $S$ is any subset of a vector space $L$, prove that $\operatorname{co} S$ is the smallest convex set containing $S$ by justifying each step in the following proof:
Let $\mathcal{C}$ be the collection of all convex sets which contain $S$ and define $B=$ $\bigcap_{C \in \mathcal{C}} C . B$ is convex, and it is the smallest convex set which contains $S$. For any sets $E \subset F \subset L$, it is always true that $\operatorname{co} E \subset \operatorname{co} F$. Therefore, $\operatorname{co} S \subset B$. But $B \subset \cos$. Hence, $\cos =B$, which proves the theorem.

Chris Trentman

Numerade Educator

02:04

Problem 3

Show that a subset $S$ of a vector space is convex iff $\alpha S+(1-\alpha) S \subset S$ for all $\alpha \in[0,1]$.

Kratika Bhadauria

Numerade Educator

Problem 4

To illustrate the operation of taking linear combinations of convex sets, let
$$
\begin{aligned}
& C_1=\left\{x \in \mathbf{R}^2 \mid x=\left(x_1, 0\right), x_1 \in[0,1]\right\} \\
& C_2=\left\{x \in \mathbf{R}^2 \mid x=\left(0, x_2\right), x_2 \in[0,1]\right\} .
\end{aligned}
$$
Let $C=\alpha C_1+(1-\alpha) C_2$. Illustrate the set $C$ for $\alpha$ equal to $1, .75$, $.5, .25$, and 0 .

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02:46

Problem 5

Let $S=\mathbf{R}_{+}^{\mathbf{n}}$. Prove that $S-S=\mathbf{R}^{\mathbf{n}}$.

Subham Jyoti Mishra

Numerade Educator

03:23

Problem 6

Let $\left(2^L\right)_0^C$ denote the set of all nonempty convex subsets of a vector space $L$. Because linear combinations of elements in this set also belong to the set, it is tempting to conclude that $\left(2^L\right)_0^C$ is a vector space. Why is this conclusion false?

Harshita Goel

Numerade Educator

00:58

Problem 7

Complete the proofs of Theorems 2.3 and 2.4.

Amit Srivastava

Numerade Educator

01:24

Problem 8

Prove the parts of Theorem 2.7 involving linear or affine subspaces.

Carson Merrill

Numerade Educator

02:11

Problem 9

Illustrate Theorem 2.7 for a map $A: \mathbf{R}^2 \rightarrow \mathbf{R}^2$.

Manik Pulyani

Numerade Educator

Problem 10

Prove Theorem 2.8.

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Problem 11

Prove that $L_{+}=\mathbf{R}_{+}^{\mathbf{n}}$ and $L_{-}=\mathbf{R}_{-}^{\mathbf{n}}$ are proper, convex, pointed cones.

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04:16

Problem 12

For $L=\mathbf{R}^{\mathbf{n}}$, define $L_{++}=\{x \in L \mid x \gg 0\}$ and $L_{--}=\{x \in L \mid$ $x \ll 0\}$. Prove that these sets are proper, convex cones. Are they pointed?

Chris Trentman

Numerade Educator

Problem 13

Verify that the set of all subsets of a set $X$ is partially ordered by $\supset$ : i.e., the definition $S \geq T$ iff $S \supset T$ yields a partial ordering $\geq$ on $2^X$. Give a simple example illustrating that this partial ordering need not be complete (usually it is not).

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08:12

Problem 14

Show that the binary relation $\geq$ on $\mathbf{R}^{\mathrm{n}}$ given by Definition 2.10 is a partial ordering and that $\mathbf{R}^{\mathbf{n}}$ equipped with that partial ordering is an ordered vector space.

Prathan Jarupoonphol

Numerade Educator

Problem 15

Verify that the function space $\operatorname{Map}(T, \mathbf{R})$ described in Chapter 1 is partially ordered by the relation $\geq$ if we define $g \geq f$ iff $g(t) \geq f(t)$ for all $t \in T$. Verify that with this partial ordering $L:=\operatorname{Map}(T, \mathbf{R})$ is an ordered vector space. Describe the positive and negative cones $L_{+}$and $L_{. .}$for this case.

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03:37

Problem 16

Find the polar cone of the following cones and illustrate both the cone and its polar in a diagram.
(a) $K=\mathbf{R}_{+}^2$
(b) $K=\left\{x \in \mathbf{R}^2 \mid x_1 \leq 0 \& x_2 \leq x_1\right\}$.

Mary Wakumoto

Numerade Educator

Problem 17

Consider a production economy with four commodities: (1) labor, (2) widgets, (3) clean shirts, and (4) soot. Activity vectors come in two varieties:
(a) A factory uses labor to produce widgets and soot using a positive scalar multiple of the activity vector $y^1=(-1,1,0,1)$.
(b) A laundry uses labor to produce clean shirts, but the labor required depends on how much soot is present. Letting $s$ denote the amount of soot, activity vectors available to the laundry conditional on the amount of soot are positive scalar multiples of $y^2=(-(s+1), 0,1,0)$.
Show by example that the aggregate technology set fails to be additive.

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Problem 18

Modify the example of a two-commodity production economy given in Section 2.2.7 to allow the first consumer to have endowment $w_1=$ $(0,1)$.

Victor Salazar

Numerade Educator

08:23

Problem 19

(a) Prove Lemmas 2.24 and 2.25 .
(b) Interpret the conclusions of both lemmas in terms of the net trade diagram.
(c) Show that Lemmas 2.24 and 2.25 reduce to Lemmas 1.10 and 1.11 when $Y=\{0\}$ (i.e., for pure exchange).
(d) Use Lemma 2.25 to provide an alternative proof of Theorem 2.23 .

Julian Gerber

Numerade Educator

00:30

Problem 20

(a) Prove Lemmas 2.27 and 2.28 .
(b) Show that these lemmas reduce to Lemmas 1.15 and 1.16 of Chapter 1 when $Y=\{0\}$.

Ankit Gupta

Numerade Educator

01:19

Problem 21

Prove Theorem 2.29 by mimicking the proof of Theorem 1.17.

Nick Johnson

Numerade Educator

02:51

Problem 22

Modify the economy described in Section 2.2.7 by assuming:
(a) the set of firms is $K=\{1,2\}$ with technology sets
$$
Y_1=Y_2=\left\{\left(y_1, y_2\right) \in \mathbf{R}^2 \mid y_1 \leq 0 ; y_2 \leq \sqrt{-y_1}\right\} ;
$$
(b) the shares of consumers in profits of the firms are given by $\theta_{11}=$ $.25, \theta_{12}=.75, \theta_{21}=.75$, and $\theta_{22}=.25$,
Solve for the Walrasian equilibrium and display the result in a net trade diagram.

Dominador Tan

Numerade Educator

02:12

Problem 23

Prove the First Fundamental Theorem of welfare economics for an Arrow-Debreu economy.

Niamat Khuda

Numerade Educator

01:22

Problem 24

Prove that a Walrasian equilibrium for a coalition production economy is Pareto optimal and in the core. Illustrate the conclusion using an appropriately modified net trade diagram.

Crystal Wang

Numerade Educator

03:27

Problem 25

Illustrate in a diagram the solution to the model of Marshallian joint supply described in Section 2.4.1 when $a_1=4$ and $a_2=6$.

Ahmad Reda

Numerade Educator

01:02

Problem 26

Modify the model of Marshallian joint supply to handle the more general case in which one sheep produces $b_1$ units of mutton and $b_2$ units of hides.

Khushbu Rani

Numerade Educator

Problem 27

Joint supply applies quite naturally to the production of contingent commodities. Consider an economy with a single consumer (Eve) and three types of commodity. Labor (commodity 3) is used to produce two contingent commodities, (1) fish when the fishing is good and (2) fish when the fish are not biting, according to the constant returns technology $Y=\left\{y \in \mathbf{R}^3 \mid y=\lambda(3,1,-1), \lambda \geq 0\right\}$. Eve is capable of supplying at most one unit of labor during the day. She maximizes the expected utility function
$$
U(x)=\pi \log \left(x_1\right)+(1-\pi) \log \left(x_2\right)
$$
where $\pi$ is the probability that fishing is good on the day in question. Eve's endowment is $w=(0,0,0)$. Normalize prices so that $p_3=1$.
(a) Find the Walrasian equilibrium allocation and prices for this economy.

Now suppose that Eve has two activity vectors to choose from, $y^1=$ $(3,1,-1)$ and $y^2=(1,2,-1)$, which can be interpreted as fishing at either of two sites. Assuming that she can split her time between the two locations, the technology set becomes
$$
Y=\left\{y \in \mathbf{R}^3 \mid y=\lambda_1 y^1+\lambda_2 y^2 ; \lambda_1, \lambda_2 \geq 0\right\} .
$$
(b) Find the Walrasian equilibrium allocation and prices for this economy. Show in a graph how the equilibrium values for $p_1, p_2, x_1$, and $x_2$ depend on $\pi$. (Hint: consider separately Case A: $\lambda_1>0, \lambda_2=0$; Case B: $\lambda_1=0, \lambda_2>0$; and Case C: $\lambda_1>0, \lambda_2>0$.)

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Problem 28

Using the parameter values of Exercise 2.25, construct a Lindahl diagram analogous to the Marshallian joint supply diagram (Figure 2.9) for the two-consumer public goods model developed in Section 2.4.2.

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